How can I fill a unitary knowing only its first column?

Question

I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a variable and solving $UU^T = I$ analytically. But this 6x6 case is out of computational reach for this method.

Is there any general method, or any clever trick, to help me fill in the rest of matrices such as $U$?

Note: I actually would prefer if all the entries were real, so technically these are better called orthogonal matrices.

$U = \frac{1}{\sqrt{5}}\begin{bmatrix} 0 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ \end{bmatrix} $

Answer 1

Martin Vesely's answer is the way to go in general, and especially if you know more than one column. However, if you're given just one column, there's an easier trick for generating a suitable unitary.

Note that $V=2|v\rangle\langle v|-I$ is a unitary ($V=V^\dagger$ and $V^2=I$). So, the question is whether you can select a $|v\rangle$ such that the first column is the one you gave ($V=U$). So, just calculate $\langle 1|V|n\rangle$ for $n=1$ to 6, and you'll find (up to a possible global phase) $$ |v\rangle=\frac{1}{\sqrt{2}}|1\rangle-\frac{1}{\sqrt{10}}\sum_{n=2}^6|n\rangle. $$ So now you can write down your unitary: $$ \frac{1}{5}\left[\begin{array}{cccccc} 0 & -\sqrt{5} & -\sqrt{5} & -\sqrt{5} & -\sqrt{5} & -\sqrt{5} \\ -\sqrt{5} & -4 & 1 & 1 & 1 & 1 \\ -\sqrt{5} & 1 & -4 & 1 & 1 & 1 \\ -\sqrt{5} & 1 & 1 & -4 & 1 & 1 \\ -\sqrt{5} & 1 & 1 & 1 & -4 & 1 \\ -\sqrt{5} & 1 & 1 & 1 & 1 & -4 \end{array}\right]. $$

Just to fill in some details: Your first column can be considered a properly normalised state $|\psi\rangle$, so you are searching for a state that satisfies $$ (2|v\rangle\langle v|-I)|1\rangle=|\psi\rangle. $$ It's easy enough to manipulate this to find (where I'm making the simplifying assumption that the inner product $\langle 1|\psi\rangle$ is real) $$ |v\rangle=\frac{|1\rangle+|\psi\rangle}{\sqrt{2(1+\langle 1|\psi\rangle)}}. $$

Answer 2

Take your vector $\frac{1}{\sqrt{5}}(0, 1, 1, 1, 1, 1)^T$ and five other arbitrary ones but at the same time these vectors have to be linearly independent. After that apply Gram-Schmidt process which produces orthonormal vectors.

Put these vectors to a matrix and you will get a unitary matrix with the first column equal to $\frac{1}{\sqrt{5}}(0, 1, 1, 1, 1, 1)^T$.